Klein bagel (4-D non-intersecting parameterization w-coordinate)


In mathematics, the Klein bottle is an example of a non-orientable surface, informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. A Klein bottle has no boundary. The “figure 8” immersion (Klein bagel) of the Klein bottle has a particularly simple parametrization. It is that of a “figure-8” torus with a 180 degree “Möbius” twist inserted. In this immersion, the self-intersection circle (when v = 0, π) is a geometric circle in the xy-plane. In four dimensions this surface can be made non-intersecting modeled after that of the flat torus. 0 ≤ θ < 2π, 0 ≤ v < 2π

Related formulas


ww-Coordinate (dimensionless)
PConstant that determine aspect ratio (dimensionless)
θThe angle that determines the rotational angle of the figure-8 as well and the position around the z-w plane (dimensionless)
ϵSmall constant related to the dependent "bump" to the fourth w axis at the intersection point (dimensionless)
vThe angle that determines the position around the figure-8 as well as the position in the x-y plane (dimensionless)